\chapter{Conclusions}
\paragraph
In this thesis, we demonstrated the feasibility of homogenization of thermal properties of porous ceramic materials using meshfree method with distance field. We showed how to combine asymptotic homogenization and meshfree method for finding thermal conductivity. We presented results of several example and investigated the effect of anisotropy in elements of thermal conductivity tensor. It is learned that the geometry of the microstructure affected the anisotropy of the thermal conductivity.
\paragraph
Since porous microstructure of the ceramic materials highly affects the macroscopic materials properties, the geometry of the microstructure has to be included into consideration during the analysis. Direct inclusion of the geometry of the microstructure into analysis at the macro level is infeasible due to excessive computational cost. That is why homogenization method was applied.
\paragraph
Homogenization requires solution of the boundary value problem at micro scale level. Mesh-based analysis methods, like Finite Element Method, require spatial meshing for solving boundary value problem. Meshing of the realistic geometry of the microstructure is difficult due to data conversion from 2D images and Computed Tomography (CT)scans. Applying meshfree method which does not need to construct the grid which must conform to the geometry, solved problems of meshing and enabled us to include realistic geometry of the microstructure to homogenization procedure.
\paragraph
Further more, dependence of the homogenized thermal conductivity to the geometry of the microstructure was investigated and following points were concluded:
\begin{itemize}
\item Porosity coefficient affects homogenized material properties
\item Rule of mixture cannot predict anisotropy of the homogenized material properties
\item Shape and distribution of the pores may result in anisotropy of the homogenized material properties
\item Orientation of the principal axes of anisotropy is well correlated with the orientation of the principal axes of the porosity structure
\item Dispersion of pores affects homogenized material properties. Higher dispersion of the pores decreases anisotropy of the thermal conductivity
\item Shape of the pores also affects the homogenized thermal conductivity
\end{itemize}
\paragraph
It should be mentioned that, the proposed approach is applicable not only to ceramic materials, but also to any kind of cellular materials such as "Metallic Foams". Furthermore, this approach is capable of predicting homogenized thermal conductivity of multiphase materials as well.
Also, The same approach could be used to determine electrical and magnetic properties of porous and multiphase materials.